Modeling and Simulation of Non-idealities in a Z-axis CMOS-MEMS Gyroscope

by

Sitaraman V. IyerB. Tech. (Indian Institute of Technology, Bombay) 1996

M. S. (Carnegie Mellon University) 1998

A dissertation submitted in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in

Electrical and Computer Engineering

Thesis Committee: Dr. Tamal Mukherjee, Chair

Prof. Gary K. FedderProf. L. Richard CarleyProf. Jonathan Cagan

Dr. Steve Bart

Department of Electrical and Computer EngineeringCarnegie Mellon University

Pittsburgh, Pennsylvania, USA

April 2003

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Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

CARNEGIE MELLON UNIVERSITY

CARNEGIE INSTITUTE OF TECHNOLOGY

THESIS

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

TITLE: MODELING AND SIMULATION OF NON-IDEALITIES IN A

Z-AXIS CMOS-MEMS GYROSCOPE

PRESENTED BY Sitaraman V. Iyer

ACCEPTED BY THE DEPARTMENT OF Electrical and Computer Engineering

MAJOR PROFESSOR DATE

DEPARTMENT HEAD DATE

APPROVED BY THE COLLEGE COUNCIL

DEAN DATE

Copyright © 2003 Sitaraman V. IyerAll Rights Reserved

To Amma and Appa

Abstract

MEMS gyroscopes have proved to be extremely difficult to manufacture reliably. The

MEMS gyroscope is required to sense picometer-scale displacements, making it sensitive

to spurious vibrations and other coupling mechanisms. This thesis aims to quantitatively

capture, through models and simulations, the sensitivity of a MEMS gyroscope to manu-

facturing variations in the widths of suspension beams and gaps between fingers in elec-

trostatic actuation and capacitive sensing combs. The gyroscope considered in this thesis

is manufactured in a CMOS-MEMS process. The suspended MEMS structures are com-

posed of the multi-layer stack of interconnect metals and dielectrics in a CMOS process.

The effect of misalignment between the metal layers in the suspended microstructures is

also modeled in the gyroscope. A number of fundamental issues related to the modeling

and simulation of MEMS gyroscopes are addressed. Models in elastic and electrostatic

domains are developed. Numerical tools such as finite element analysis or boundary ele-

ment analysis are used for model verification. Behavioral simulation is used throughout

this thesis to analyze the gyroscope and system-level design issues.

The elastic modeling effort is primarily aimed at a thorough understanding of cross-

axis coupling in micromechanical springs and at multi-dimensional curvature in the multi-

layer suspended structures in the CMOS-MEMS process. Cross-axis stiffness constants

are derived for basic spring topologies such as crab-leg, u-spring and serpentine springs.

Techniques to reduce, and even completely eliminate, elastic cross-axis coupling are dis-

cussed. In the electrostatic domain, a methodology which combines analytical equations

with numerically obtained data is developed to model CMOS-MEMS combs. Particular

attention is paid in this methodology to make the resultant behavioral model energy con-

serving. Convergence problems found in behavioral simulations of gyroscopes lead to a

detailed comparison of different Analog Hardware Description Language (AHDL) model

implementation of mechanical second-order systems, such as the resonating structure in a

gyroscope. AHDL model implementation guidelines for improved convergence in behav-

ioral simulations are deduced from the comparisons.

ix

Using the elastic and electrostatic models as the basis, analytical equations relating

gyroscope non-idealities: the Zero Rate Output, acceleration and acceleration-squared

sensitivity and cross-axis sensitivity to manufacturing effects are derived. The equations

are compared with results of behavioral simulation. Monte Carlo simulations using the

behavioral models are run in order to verify the trends predicted by the analytical equa-

tions. The analysis and simulations result in several insights into gyroscope non-idealities

and design pointers to reduce them.

x

Acknowledgments

There are a number of individuals who have contributed to the completion of this the-

sis, technically as well as philosophically. First, I would like to thank my advisor Dr.

Tamal Mukherjee. In the course of my graduate studies I have benefitted from his experi-

ence in a number of aspects of conducting scientific research. In addition to the numerous

technical discussions we had, which have shaped this thesis considerably, his feedback on

documents and presentations have vastly improved my communication skills. His

patience, encouragement and dedicated mentoring of his students is, in my opinion,

extraordinary, if not uncommon.

I would also like to thank Prof. Gary Fedder whose grasp of the physics of MEMS has

aided my comprehension, particularly, in developing elastic models. His considerable

experience in MEMS inertial sensors has also provided a better perspective on many occa-

sions. Technical feedback from Prof. Rick Carley, Prof. Jon Cagan and Dr. Steve Bart has

helped me set a broader context for this thesis, and is also greatly acknowledged. Prof.

Wojciech Maly and Prof. Jim Hoburg were also gracious in sharing their expertise and

helped in guiding the direction of the thesis.

Simulations and measurements done were greatly expedited due to colleagues from

the MEMS laboratory at Carnegie Mellon as well as the Center for Silicon System Imple-

mentation. My latest officemates Bikram Baidya, Kai He and Qi Jing have helped me on

numerous occasions. In particular, Bikram with extraction and FEM simulations, Qi Jing

with several discussions on NODAS, and Kai with support for layouts and CAD. I would

also like to acknowledge Hasnain Lakdawala for his collaboration with a number of mea-

surements, modeling discussions and his general opinion on the state of the art, Xu Zhu

for releasing the chips used for measurements, Hao Luo for sharing his gyroscope design

for use in this thesis and Huikai Xie for discussions on gyroscope measurements. Mary

Moore and Drew Danielson in the MEMS Laboratory, Roxann Martin in the Center for

Silicon System Implementation and Elaine Lawrence and Lynn Philibin in the graduate

office have helped in sorting out procedural and administrative issues on numerous occa-

xi

sions. Their friendly countenance made them very approachable and things were that

much smoother with them around.

Financial support, under various grants from the Defense Advanced Research

Projects Agency (DARPA), the National Science Foundation (NSF) and the MARCO

Center for Circuits, Software and Systems is gratefully acknowledged.

A number of different officemates at different times made courses, research and grad-

uate student life all the more enjoyable. Additionally, friends in CMU and Pittsburgh have

made the sometimes tedious and bumpy graduate school, a pleasant experience. All of

them have left their positive impressions on me. Kumar Dwarakanath, Vikram Talada,

Hasnain Lakdawala, Sanjay Rao, Sudha Dorairaj, Madhavi Vuppalapati, Bikram Baidya,

Debanjali Dasgupta and many others have kept my spirits aloft. Jam sessions with Aneesh

Koorapaty and Sundar Vedula were also immensely enjoyable and always lightened the

pressure.

I would like to acknowledge the completely selfless and unrelenting support of my

two sisters Shanthi and Vidya, their husbands and my nephew and niece. Their high

expectations and confidence in me, have pushed me farther than I would have ever

attempted. Some of the most difficult times in the last couple of years would have been

unthinkable without them. My wife Vidhya, whose cheerfulness and wisdom has made

some very stressful times bearable, has been a great companion in the past few months.

Finally, it should be mentioned that, the sheer will power of my mother and the excellent

engineering mind of my father, have not only aided and guided me in my career, but have

also set me high standards to emulate. These and many of their extraordinary qualities

have carried me thus far and will continue to do so.

xii

Table of Contents

Abstract ........................................................................................ ix

Acknowledgments ....................................................................... xi

Table of Contents ...................................................................... xiii

List of Tables ........................................................................... xviii

List of Figures ............................................................................. xx

Chapter 1. Introduction.................................................................................... 1

1.1 Introduction..................................................................................................1

1.2 Motivation....................................................................................................2

1.3 Scope of This Thesis....................................................................................5

1.4 Thesis Organization .....................................................................................6

Chapter 2. Background.................................................................................... 7

2.1 Introduction..................................................................................................7

2.2 Micromachined Gyroscopes ........................................................................72.2.1 Brief History..........................................................................................................7

2.2.2 Common Features of Surface-Micromachined Vertical Axis Gyroscopes............8

2.2.3 Vertical Axis CMOS-MEMS Gyroscope ..............................................................9

2.3 CMU CMOS-MEMS Process....................................................................10

2.4 Structured Design Methodology for MEMS..............................................132.4.1 Modeling .............................................................................................................15

2.4.2 Elastic Models .....................................................................................................16

2.4.3 Electrostatic Modeling.........................................................................................19

2.5 Summary....................................................................................................20

Chapter 3. Elastic Modeling .......................................................................... 21

3.1 Introduction................................................................................................21

3.2 Stiffness Matrices.......................................................................................23

3.3 Modeling....................................................................................................24

3.4 Model Verification .....................................................................................31

3.5 Accelerometer Simulation .........................................................................36

3.6 Symmetric springs .....................................................................................37

xiii

3.7 In-plane to Out-of-plane Elastic Cross-axis coupling................................433.7.1 Rotation of Principal Axes in Multi-layer Beam ................................................ 44

3.8 Geometrical Interpretation of Cross-axis Coupling...................................50

3.9 Manufacturing Variations and Elastic Cross-axis Coupling ......................53

3.10 Curl Modeling............................................................................................583.10.1 Extension of Multimorph Analysis ..................................................................... 59

3.10.2 FEA Verification ................................................................................................. 63

3.10.3 Measurements ..................................................................................................... 65

3.11 Summary ....................................................................................................67

Chapter 4. Reduced Order Models: From Beams to Springs .....................69

4.1 Introduction................................................................................................69

4.2 Spring Stiffness Computation ....................................................................704.2.1 Background......................................................................................................... 70

4.2.2 Stiffness Computation Procedure ....................................................................... 71

4.3 Translation from Circuit-level to Functional Schematic............................744.3.1 Nested gyroscope................................................................................................ 74

4.3.2 Z-axis accelerometer ........................................................................................... 76

4.4 Verification.................................................................................................774.4.1 Verification of Spring Stiffness Computation..................................................... 77

4.5 Comparison of Circuit-level Schematic with Functional Schematic

Simulation ..................................................................................................794.5.1 Example 1: Nested-Gyroscope Design Space Exploration ................................ 79

4.5.2 Example 2: Resonance Frequency Analysis of Z-axis Accelerometer ............... 80

4.6 Inertial and Viscous Effects and Extension to Arbitrary Spring Topologies .

814.6.1 Model Formulation ............................................................................................. 81

4.6.2 Model-order Reduction....................................................................................... 84

4.6.3 General Reduced-order Modeling ...................................................................... 85

4.7 Summary ....................................................................................................86

Chapter 5. Electrostatic Modeling of CMOS-MEMS Comb..........................87

5.1 Introduction................................................................................................87

5.2 Background ................................................................................................89

5.3 Modeling Goals and Approach ..................................................................91

5.4 Analytical Model for Movement in Gap Direction ...................................91

5.5 Design of Experiments and Simulation .....................................................965.5.1 Comb Parameterization and Variable Selection ..................................................96

5.5.2 Variable Screening...............................................................................................97

5.5.3 Choice of Variable Ranges ..................................................................................97

5.5.4 Data Collection....................................................................................................99

5.6 Modeling Methodology ...........................................................................1015.6.1 Capacitance Modeling .......................................................................................101

5.6.2 Combined Capacitance-Force Modeling ...........................................................103

5.7 Differential Comb Modeling....................................................................104

5.8 BEA Summary.........................................................................................107

5.9 Results......................................................................................................1075.9.1 Model Generation by Curve Fitting ..................................................................107

5.9.2 Behavioral Model Implementation....................................................................109

5.10 Experimental Verification ........................................................................111

5.11 Summary..................................................................................................112

Chapter 6. Convergence and Speed Issues in MEMS Behavioral

Simulation..............................................................................................114

6.1 Introduction..............................................................................................114

6.2 Background..............................................................................................1156.2.1 Numerical Integration........................................................................................116

6.2.2 Time Discretization of Components..................................................................117

6.3 Model Formulation ..................................................................................1196.3.1 Multi-domain Simulation ..................................................................................120

6.3.2 Implementation of Time-derivatives .................................................................121

6.3.3 Displacement as across variable ........................................................................122

6.3.4 Velocity as across variable.................................................................................124

6.4 Simulation Results ...................................................................................125

6.5 Model Implementation Example: Squeeze-film Damping ......................127

6.6 Summary..................................................................................................129

Chapter 7. Analysis and Simulation of a CMOS-MEMS Gyroscope........ 131

7.1 Introduction..............................................................................................131

7.2 Gyroscope Description and Circuit-level Representation .......................133

xv

7.2.1 Gyroscope Description ..................................................................................... 133

7.2.2 Gyroscope Parameters ...................................................................................... 136

7.2.3 Notation ............................................................................................................ 137

7.2.4 CMOS-MEMS Gyroscope Design Parameters................................................. 139

7.3 Gyroscope Sensitivity ..............................................................................141

7.4 Effect of Asymmetrical Drive..................................................................148

7.5 Zero Rate Output (ZRO)..........................................................................1527.5.1 Beam Width Variation....................................................................................... 153

7.5.2 Comb gap variation........................................................................................... 159

7.5.3 Mask Misalignment .......................................................................................... 165

7.5.4 ZRO Summary .................................................................................................. 168

7.6 Acceleration Sensitivity ...........................................................................1687.6.1 Beam Width Variation....................................................................................... 170

7.6.2 Comb Gap Variation ......................................................................................... 171

7.6.3 Mask Misalignment .......................................................................................... 173

7.6.4 Summary of Acceleration Sensitivity ............................................................... 174

7.7 Cross-axis Sensitivity ..............................................................................1747.7.1 Rotation About Sense Axis ............................................................................... 175

7.7.2 Rotation About Drive Axis () ........................................................................... 177

7.7.3 Summary of Cross-axis Sensitivity................................................................... 178

7.8 Simulation Results ...................................................................................1787.8.1 Mismatch Simulation Results ........................................................................... 179

7.8.2 Monte-Carlo Simulations.................................................................................. 181

7.9 Summary ..................................................................................................185

Chapter 8. Summary and Future Work .......................................................186

8.1 Thesis Summary and Contributions.........................................................1868.1.1 Modeling ........................................................................................................... 186

8.1.2 Simulation ......................................................................................................... 187

8.1.3 Gyroscope Design............................................................................................. 187

8.2 Future Directions of Work .......................................................................188

References .................................................................................190

Appendix A1 Mathematica program to derive Crab-leg stiffness matrix ..199

Appendix A2 Equations for out-of-plane off-diagonal stiffness constants ....

202

Appendix A3 Mesh Refinement Steps ......................................................... 206

Appendix A4 Comb Model Listing ............................................................... 213

Appendix A5 Comparison of Rotational and Translational Modal Frequen-

cies .......................................................................................... 238

Appendix A6 Beam Widths and Comb Gaps for Monte-Carlo Analysis .. 240

Appendix A7 OCEAN Scripts and Sample Netlist for Gyroscope Monte-Car-

lo Simulations ......................................................................... 242

A7.1 OCEAN Script .........................................................................................242

A7.2 Gyroscope 2D netlist ...............................................................................244

xvii

List of Tables

3.1 Comparison of FEA and analytical stiffness (in-plane) values for the crab-leg

spring........................................................................................................................32

3.2 Comparison of FEA and analytical stiffness (in-plane) values for the u-spring ......33

3.3 Comparison of FEA and analytical stiffness (in-plane) values for the serpentine

spring........................................................................................................................33

3.4 Comparison of kyz from NODAS and FEA.............................................................50

3.5 Vertical deflection (in µm) of tip in 100 µm: Macromodel vs. FEA ......................64

3.6 Vertical deflection (in µm) for simplified accelerometer .........................................65

5.1 Comparison of beam and comb models ...................................................................89

5.2 Mapping of points from Z plane to W plane............................................................94

5.3 Experimental plan set 1 for simple comb BEA........................................................97

5.4 Experimental plan set 2 for simple comb BEA........................................................98

5.5 Experimental plan to obtain Fx values for simple comb using BEA .....................101

5.6 Experimental plan set 1 for differential comb BEA...............................................105

5.7 Experimental plan set 2 for differential comb BEA...............................................105

5.8 Points to be omitted due to comb finger crashing..................................................106

5.9 Summary of BEA runs for simple comb................................................................107

6.1 Typical ranges for various physical domains in MEMS ........................................120

6.2 Comparison of five implementations .....................................................................125

7.1 Symbols used..........................................................................................................138

7.2 Geometrical parameters of the CMOS-MEMS gyroscope ....................................139

7.3 Functional parameters of the CMOS-MEMS gyroscope .......................................140

7.4 ZRO resulting from linear gradients in beam width ..............................................153

7.5 Comparison between analytical calculations and NODAS simulations of ZRO...179

7.6 Comparison between analytical calculations and NODAS simulations of

acceleration sensitivity ...........................................................................................180

xviii

7.7 Comparison between analytical calculations and NODAS simulations of cross-axis

sensitivity ...............................................................................................................180

A2.1 Comparison of FEA and analytical stiffness (out-of-plane) values for the crab-leg

spring......................................................................................................................204

A2.2 Comparison of FEA and analytical stiffness (out-of-plane) values for the U spring...

204

A2.3 Comparison of FEA and analytical stiffness (out-of-plane) values for the serpentine

spring......................................................................................................................205

A6.1 Fractional variation of beam widths and gaps used for Monte-Carlo simulations.240

List of Figures

1.1. Working principle of a microgyroscope with a sensing accelerometer nested insidea vibrating frame. .......................................................................................................2

1.2. Cross-section of microstructures in a CMOS-MEMS process...................................4

2.1. Topology comparison of (a) single-layer surface micromachined gyroscopes (e.g.,Clark et al. [19]) and (b) CMOS-MEMS nested gyroscope [10]. The dark shadedcombs are the drive combs. Shading in the sense combs indicates differentpotentials. ...................................................................................................................8

2.2. (a) SEM of the vertical axis CMOS-MEMS nested-gyroscope [10]. (b) Functionallyequivalent structure showing the inner accelerometer, outer rigid frame, inner andouter springs and drive and sense combs. ................................................................10

2.3. Abbreviated process flow for post-CMOS micromachining developed at CMU[7][31][35] (a) CMOS wafer cross-section with circuits and interconnects (soon tobe microstructures) (b) Oxide removal step (c) Microstructure release by Siliconremoval.....................................................................................................................11

2.4. (a) Lateral curling seen in beams with deliberately misaligned metal layout (b)Cross-section of the beam. It is seen that the METAL3 is not aligned with theMETAL2 and METAL1. (Pictures courtesy Xu Zhu and Hasnain Lakdawala) ......12

2.5. (a) Geometrical offset in a differential comb-drive used in a CMOS accelerometer.One of the gaps is smaller than the other one (b) Laterally curled springs in anaccelerometer (Pictures courtesy Vishal Gupta) ......................................................13

2.6. MEMS design hierarchy...........................................................................................14

2.7. The direction along the length (x) and the direction of deflection of a beam (y).....16

2.8. (a) Spring with single-chain of 9 beams attached to a plate. C is the point ofapplication of force. The other end of the spring is anchored. (b) Free-body diagramof beam 6 and the bending moment along beam 6...................................................18

3.1. Outer frame of a gyroscope driven by a sinusoidal voltage source and a DC source.Motion of point C is shown on the right. (a) With ideal springs oscillations are onlyalong the y direction. (b) However, with non-ideal mismatched springs smallamount of motion couples to the x direction............................................................21

3.2. Elements of the stiffness matrix and the in-plane and out-of-plane sub-matricesand . This symmetric matrix has 21 distinct terms. If the shaded elements are zero,the number of distinct non-zero terms reduces to 12. ..............................................23

xx

3.3. Design variables for crab-leg-spring, U-spring and serpentine spring with proof-mass. The external forces and moments are applied at C, the centroid of the plate,with only one spring in the analysis so that all the cross-axis terms can be clearlyobserved. ..................................................................................................................25

3.4. Forces and moments applied at the centroid of a proof-mass attached to the free endof a crab-leg. Boundary conditions are applied as equality constraints on the threedisplacements. ..........................................................................................................26

3.5. Trends in the variation of in-plane spring constants for the U-spring for varyingbeam lengths (Lb1). The design variables are set to: w=2.0 µm, Lt=10.0 µm,Lb2=200.0 µm...........................................................................................................30

3.6. Trends in the variation of in-plane spring constants for the serpentine-spring forvarying beam lengths (a). The design variables are set to: w=2.0 µm, Lb=20.0 µm,n=4............................................................................................................................31

3.7. Comparison of analytical model and FEA for crab-leg-spring kxy for varying crab-leg thigh lengths (Lt). The design variables are set to: w=2.0 µm, Ls=50.0 µm ......35

3.8. Comparison of analytical model and FEA for u-spring kxy for varying U-springbeam lengths (Lb1). The design variables are set to: w=2.0 µm, Lt=10.0 µm,Lb2=Lb1-30.0 µm ......................................................................................................35

3.9. Comparison of analytical model and FEA for serpentine-spring kxy for varyingserpentine-spring beam lengths (a). The design variables are set to: w=2.0 µm,Lb=20.0 µm, n=4 ......................................................................................................36

3.10. NODAS simulation of cross-axis sensitivity in y-accelerometer. The structure ofthe accelerometer with four serpentine springs is shown on the side. .....................37

3.11. Axes of symmetry for the u-spring and serpentine spring .......................................38

3.12. Example of a symmetric spring with the axes of symmetry along the y direction.The load point C, the anchor point A and the end points of beam number 9 are alsoshown. ......................................................................................................................40

3.13. Rotation of beam principal axes due to asymmetrical cross-section: asymmetricalside-walls in the single-layer case and misaligned metal layers in the multi-layercase ...........................................................................................................................44

3.14. (a) Side view of a n-layer beam of length L (b) Cross-section of the beam with dotsrepresenting the axial forces acting out of plane......................................................45

3.15. Cross-section of beams used for comparison of macromodel with FEA. The kyzmatches to 2% for 2 sets (50 µm X 3.0 µm and 100 µm X 2.1 µm) of 3 beams each.49

3.16. In-plane rotation and displacement of the principal axes of elasticity .....................51

3.17. Signs of off-diagonal elements in the in-plane stiffness matrix ...............................53

3.18. Linear variation in beam-widths across a wafer, mapped onto the springs of a singledevice with a plate suspended by four springs .........................................................54

3.19. (a) Model of a CMOS cantilever beam composed of metal, dielectric andpolysilicon layers (b) Cross-section of an asymmetric multi-layer beam with dotsrepresenting the axial forces acting out of plane. Since the forces areasymmetrically located there is a resultant lateral bending moment in addition tothe vertical bending moment ....................................................................................59

3.20. Norton equivalent of a beam macromodel with thermally induced lumped forceand moment sources and an embedded piezoresistor. The beam has threetranslational pins, three rotational pins and one electrical pin at each port. ............62

3.21. (a) Cross-section of beams. Metal3 is 2.1 µm wide and Metal2 and Metal1 layersare 1.8 µm wide and are offset by -0.15 µm from the center of the beam (b)Comparison of behavioral curl model with FEA for beam of length 100 µm.Difference is less than 3% for all temperatures........................................................64

3.22. Temperature-induced curling of a simplified accelerometer structure obtained from3D FEA. The vertical deflections at points (a) and (b) are compared with resultsfrom behavioral simulation using the macromodel in Table 3.6. .............................65

3.23. (a) SEM of the test structure used to characterize beam curling with temperature. Itconsists of alternating misaligned and symmetric beams (b) Cross-section of themeasured beams (misaligned) ..................................................................................66

3.24. Interferometric images of the out-of-plane curl of the test structure at 24oC and56oC. One fringe length corresponds to 245 nm displacement in the verticaldirection....................................................................................................................67

3.25. Comparison of relative tip deflection from measurements and macromodel showinga match to within 15%. 100 nm and 20 nm error bars are shown for the z and xdeflections respectively. ...........................................................................................67

4.1. (a) Layout of a nested-resonator system (b) Corresponding NODAS schematicobtained through layout extraction. The schematic consists of a central plateconnected through the four inner springs to the frame. The frame is composed of

xxii

four plates which are suspended by the four outer springs. The other ends of thefour outer springs are connected to the chip substrate. ............................................75

4.2. Functional model generated from the circuit-level schematic of the nested-resonatorsystem shown in Figure 4.1(b) .................................................................................75

4.3. (a) Layout of a spring with about 50 beams connected to a proof-mass at one endand anchored at the other end (b) Corresponding NODAS schematic of the springobtained through layout extraction. Beams marked with “1” and “2” have widthsw1 and w2 respectively. ...........................................................................................76

4.4. Functional schematic generated from the circuit-level schematic of theaccelerometer shown in Figure 4.3(b)......................................................................77

4.5. Layout of the spring used for FEA. A is the anchored point. M is the point to whichthe mass is attached. The length l and the width w of the vertical beam are variedover a range of values...............................................................................................78

4.6. Comparison of spring stiffness computation for kxx with FEA................................78

4.7. Comparison of spring stiffness computation for kyy with FEA................................78

4.8. Comparison of spring stiffness computation for with FEA ..........................78

4.9. Difference in the resonant frequency extracted from the schematic simulations andbehavioral simulations (a) drive-mode (b) sense-mode ...........................................79

5.1. (a) Top view of a simple comb with three comb fingers (the lesser of the twonumbers is taken as the number of fingers in the comb) (b) Cross-section of a combfinger in the CMOS-MEMS process........................................................................87

5.2. Top view of a differential comb along with the equivalent capacitive dividerschematic. .................................................................................................................88

5.3. (a) Cross-section of a comb showing two fixed fingers (F) and one movable finger(M) along with odd and even symmetry planes used in derivation of conformal-mapping based analytical models. (b) Movement in the gap direction breaks thesymmetry..................................................................................................................89

5.4. Simplification of laterally displaced comb cross-section using symmetry (a) Combsection showing 3 fixed fingers and 2 movable fingers displaced in the x directionfrom the nominal symmetrical position (shown with dotted lines). (b)Simplification by introduction of odd symmetry planes (c) Equivalent configurationwith the fixed comb-fingers replaced by odd-symmetry plane placed midwaybetween the rotor and stator fingers .........................................................................92

kφzφz

5.5. (a) Conformal mapping for a single conductor placed asymmetrically between twoground planes ...........................................................................................................93

5.6. Comparison of analytical model adapted from [75] and FEA for x displacement ofthe comb-finger cross-section shown in Figure 5.4. ................................................95

5.7. Boundary-element mesh for a 10 finger vertically curled lateral comb with allCMOS layers. .........................................................................................................100

5.8. Histogram of error in % between model and numerical data for 4566 capacitancepoints. The error is almost within ± 3% .................................................................108

5.9. Histogram of error in % between model and Fx from numerical derivative ofcapacitance w.r.t x...................................................................................................108

5.10. Histogram of error in % between model and Fy obtained from numerical derivativeof capacitance w.r.t. y .............................................................................................108

5.11. Histogram of error in % between model and Fz obtained from numerical derivativeof capacitance w.r.t. z .............................................................................................108

5.12. Histogram of capacitance error in % between model and numerical data for thedifferential comb ....................................................................................................109

5.13. Functional diagram of the CMOS-MEMS gyroscope used in simulation [10]......110

5.14. Interferometry image showing the vertical offset in drive comb ...........................110

5.15. Drive amplitude variation of a gyroscope with change in vertical overlap betweenthe movable and fixed portions of the comb actuator ............................................110

5.16. (a) SEM of capacitance test structure with in-built heaters. The curling can bechanged by changing the current passing through the polysilicon wires which passthrough the outer frame of the structure. Interferometry images of a quarter of thestructure at (b) room temperature and (c) heated are also shown. .........................111

5.17. Capacitance change measurement schematic shown with heating resistors for eachstructure. Cp is the parasitic capacitor whose value is obtained using layoutextraction................................................................................................................112

5.18. Comparison of measured and predicted capacitance change. The two sets ofmeasured data correspond to the voltage applied to each of the two resistors, with0.1 V applied to the other resistor. .........................................................................112

6.1. Computation of state variable using Backward Euler integration rule. The areas ofthe rectangles obtained by integration are shown. .................................................117

xxiv

6.2. Circuit interpretation of Backward Euler integration rule for (a) Parallel RLCnetwork (b) Mechanical spring-mass-damper system modeled by use of twoadditional states to hold vn and an ..........................................................................118

6.3. In-plane stiffness matrix for a beam [61]. Beam with length = 100 µm, width = 2µm, thickness = 2 µm and Young’s Modulus E = 165 GPa. A is the cross-sectionarea and I is the moment of inertia. (a) The large span of the diagonal elements ofthe stiffness matrix is evident. (b) The stiffness matrix after scaling the rotationaldiscipline by has much smaller condition number. ...............................................121

6.4. Squeeze-film damping modeled by equivalent resistors and inductors. Only the firsttwo R-L branches are shown in the figure. More accurate models need morenumber of branches. ...............................................................................................128

6.5. VerilogA code and equivalent circuit for first implementation of the squeeze-filmdamping model. Only two of the RL branches are shown. The actual circuitinterpretation by the simulator is not exactly known but is probably more complexbecause it is observed that additional states are implicitly introduced duringsimulation. ..............................................................................................................128

6.6. VerilogA code and equivalent circuit for second implementation of the squeeze-film damping model. The controlled sources inside the damping model are morelocally distributed compared to the first implementation. This implementationshows better convergence properties......................................................................129

7.1. (a) Nested gyroscope design showing the drive and sense combs, the outer andinner springs, the input axis, the direction of driven vibrations and the direction ofCoriolis-force induced (sense) vibrations (b) Sense capacitance bridge formovement of inner proof-mass in positive x axis ..................................................131

7.2. (a) SEM of the nested-gyroscope (b) Corresponding NODAS schematic obtainedthrough layout extraction. ......................................................................................134

7.3. (a) Output spectrum of an ideal gyroscope for an input sinusoidal rotation rate (b)sense schematic showing demodulation of gyroscope capacitance bridge output toyield voltage proportional to input rate. The angle needs to be adjusted tomaximize sensitivity and minimize offsets ............................................................135

7.4. Classification of sources of microgyroscope non-idealities...................................137

7.5. Anti-phase voltages applied to drive the gyroscope into oscillations. ...................142

7.6. Nested resonator system and dynamical equations (a) when a force Fco is applied tothe outer frame and (b) when a force Fci is applied to the inner mass. ..................145

7.7. Asymmetrical drive: Actuation voltage applied to only the top linear comb ........149

7.8. Example case for beam width variation and equation for coupling, w1 = w2 = w3≠w4............................................................................................................................154

7.9. Example case for beam width variation, w1 = w2 = w3≠ w4. .................................157

7.10. Mismatch in the gaps in the drive combs on the top and bottom...........................159

7.11. Mismatch in the gaps in the sense combs on the two sides....................................163

7.12. Cross-section of one set of fingers of a differential sense comb (a) without lateraloffset, vertical motion leads to common-mode capacitance change; (b) with lateraloffset, vertical motion leads to common-mode and differential capacitance change. .165

7.13. Spectrum of output voltage of a non-ideal gyroscope when subjected to an externalacceleration.............................................................................................................169

7.14. ZRO Histograms for widths and gaps = 1.8 µm and 2.0 µm from Monte-Carlosimulation. ..............................................................................................................182

7.15. Acceleration sensitivity histograms for widths and gaps = 1.8 µm and 2.0 µm fromMonte-Carlo simulation. ........................................................................................182

7.16. Acceleration-squared sensitivity histograms for widths and gaps = 1.8 µm and 2.0µm from Monte-Carlo simulation. .........................................................................183

7.17. Trade-off between gyroscope sensitivity and acceleration sensitivity with varyinggap ..........................................................................................................................184

A3.1. Regions defined in the boundary element model for mesh refinement. REFINE0 isthe most finely meshed region, followed by REFINE1 and REFINE3 and finallyREFINE2 and REFINE4 are the most coarsely meshed regions. ..........................206

A5.1. Rectangular plate suspended by four springs .........................................................238

xxvi

Chapter 1. Introduction

1.1 IntroductionThe field of Microelectromechanical Systems (MEMS) has, over the past 20 years,

emerged as a technology that promises to have significant impact on everyday living in the

near future. MEMS provide inexpensive means to sense and, in a limited way, control

physical, chemical and biological interactions with nature. They add a new dimension to

the information revolution of the latter half of the twentieth century, by enabling ubiqui-

tous access to sensor data previously limited to industrial, military, and medical applica-

tions. MEMS seek to achieve this vision through a variety of manufacturing techniques

common among which are surface micromachining, bulk micromachining and LIGA [1].

These integrated circuit (IC)-like techniques are capable of producing micrometer-scale

features. However, they lack the precision (i.e., relative accuracy) of traditional mechani-

cal fabrication practices. Being integrated circuit compatible, they derive their power by

leveraging well-understood and characterized signal processing capabilities of integrated

circuits. As a result, a wide spectrum (literally as well) of applications have been made

possible such as inertial sensors, pressure and acoustic transducers, high frequency radios,

optical communications, lab-on-a-chip for chemical and biological analysis.

The contributions of this thesis are primarily relevant for micromachined inertial sen-

sors. Accelerometers and gyroscopes are two important members of the inertial sensor

family. Accelerometers sense the external acceleration in which they are placed, while

gyroscopes measure the rate of rotation or the angular velocity of the object to which they

are attached. Multi-axial accelerometers and gyroscopes can be combined to build an Iner-

tial Measurement Unit (IMU), also called an Inertial Navigation System (INS). Tradition-

ally, high precision IMUs have been an indispensable part of ships, aeroplanes, satellites,

space shuttles and the like. Surface micromachined inertial sensors, which can be batch-

fabricated with low cost have a small sensing proof-mass (~ micrograms) and conse-

quently lower resolution compared with macro-scale accelerometers or optical gyroscopes

[2]. The availability of low cost inertial sensors has opened up a wide range of new appli-

cations which do not require the high precision that IMUs demand. The current market for

1

inertial sensors in automobiles is estimated to be about a billion dollars per year [3]. Air-

bag-deployment in automobiles is a well known example of a commercially successful

low cost, low resolution application. Surface-micromachined gyroscopes, have applica-

tions in dynamic stability control and rollover detection in automobiles, computer mice,

pointers, video camera stabilization and a number of robotics and military applications

[2][4]. Conventional rotating-wheel gyros and high-precision fiber-optic and ring laser

gyros are too expensive and too large to be adopted into the market for micro gyroscopes

[2]. While potential markets for inexpensive gyroscopes exist, technical challenges have

been impeding the rapid commercial deployment of gyroscopes. In the next section some

of the fundamental problems that have been encountered in manufacturing robust micro-

machined gyroscopes are examined.

1.2 MotivationMicrogyroscopes are mainly attractive because of their small size (~ 1 mm X 1 mm

including sensing circuits) and low cost. Most microgyroscopes consist of a vibrating

proof-mass which is driven into oscillation by electrostatic or other means. When placed

in a rotational field, the vibrating proof-mass experiences an apparent force called the

Coriolis force, which is proportional to the cross-product of the angular velocity of the

rotational field and the translational velocity of the oscillating proof-mass (Figure 1.1).

The Coriolis force is orthogonal to the direction of the driven oscillation. The displace-

ment induced by the Coriolis force is picked up by a sense accelerometer, which can either

FIGURE 1.1. Working principle of a microgyroscope with a sensing accelerometernested inside a vibrating frame.

Fc 2MsΩ vd×=

GΩΩ

R

Cv

F

Ms

Md

Bd

KdKsBs

vd

Driven oscillations

Inducedoscillations

Fc

Coriolis Force

Inner accelerometer

2

utilize the vibrating proof-mass or have a separate sense proof-mass. Figure 1.1 shows the

microgyroscope chip C which is attached to a rotating frame R. The global inertial refer-

ence frame G is also shown. The velocity of driven oscillation and Coriolis force Fc are

shown in mutually orthogonal directions. The zoomed in view of the gyroscope schemati-

cally shows an inner accelerometer nested inside an outer resonator. The outer accelerom-

eter is driven into oscillation. Orthogonal induced oscillations between the two are picked

up by a capacitive sensing circuit. In order to identify important design issues, a high-level

analysis of typical magnitudes of various microgyroscope quantities is presented below.

In the following analysis typical numbers for microgyroscopes are used in order to

bring out the relative magnitudes of displacements and velocities in the driven (oscilla-

tion) direction and the induced (Coriolis force) direction. Typical value of the sense mass

is about . The angular velocities that can be sensed are of the order of

. The oscillations are usually about 10 kHz with an amplitude of about 5 µm.

Therefore, the peak oscillation velocity is about . The Coriolis force is

then given by . Assuming a spring stiffness for the sense

accelerometer of , the sense displacement is about 10 pm. In any real microgyro-

scope, some part of the driven oscillation couples onto the sense accelerometer, through

electrostatic, inertial, viscous and elastic modes. Comparing the magnitudes of the driven

oscillation and the displacement produced by the sense accelerometer, it is seen that

undesired coupling from the driven oscillation to the sense oscillation should be as small

as 2 ppm. While this may be a difficult number to achieve in any low cost system, it is

almost impossible to realize such precise dimension-control in IC-based processes which

typically control relative fabrication tolerances to only about 1% or 10000 ppm [5]. Fur-

thermore, in capacitive sensors, a displacement of few picometers typically results in a

capacitance change of a few zepto farads ( ). The total sense capacitance and para-

sitic capacitances are usually of the order of ten to hundred femto farads, leading to a rela-

tive capacitance change of 0.1 ppm. Therefore, extremely low noise front ends are

required to sense such small relative capacitance changes. Another fundamental issue

which limits the resolution of both microaccelerometers and microgyroscopes is mechani-

vd

Ms 1µg=

Ω 1° s⁄=

vd 0.31m s⁄=

Fc 2MsΩ vd× 10pN= =

1 N m⁄( )

10 21–

3

cal thermal noise. Since IC processes and surface micromachining are both basically thin

film processes, the resulting proof-masses tend to have a large surface-area to volume

ratio. As a result, viscous damping forces are more significant at the micro-scale than at

the macro-scale. Therefore, the Brownian motion resulting from viscous loss mechanisms

sets a lower limit on the smallest deterministic motion that can be sensed [6].

The solutions to the above fundamental challenges need a multi-pronged approach.

The CMOS-MEMS process developed at Carnegie Mellon provides partial answers to

some of the challenges [7] (Figure 1.2). Tight integration of MEMS and sensing circuits

leads to minimized parasitic capacitances. Large gaps between the MEMS structure and

the substrate (~ 30 µm) lead to reduced Couette damping on the underside of the structure.

In addition, the CMOS-MEMS process has several other advantages including full com-

patibility with a standard CMOS process, 0 additional masks for MEMS processing, high

aspect ratio MEMS structure and multi-conductor stacks which facilitate complex routing.

However, the CMOS-MEMS process also has a few inherent limitations such as vertical

curling due to the multi-layer structures, inadequate control of the beam cross-sections and

lack of control over mechanical properties of the microstructure.

FIGURE 1.2. Cross-section of microstructures in a CMOS-MEMS process

METAL3

METAL2METAL1POLY

OXIDE

Si

ReleasedMicrostructures

CMOS Circuits

4

1.3 Scope of This ThesisThe primary goal of this thesis is to understand the effect of elastic and electrostatic

coupling on a CMOS-MEMS microgyroscope. Specifically, the microgyro parameters

under consideration are [8][9]:

• Zero Rate Output (Input offset or Bias): Input rate required to drive the output volt-

age to zero.

• Acceleration and Acceleration-squared Sensitivity: A linear acceleration applied to

the gyroscope may result in an output indistinguishable from that produced by an input

rotation. Typically, gyroscopes show a linear as well as quadratic dependence on accel-

eration.

• Cross-axis Sensitivity: Output produced by an angular rotation about an axis orthogo-

nal to the input axis of the gyroscope.

Little or no attention has been paid to the above parameters in public literature and

therefore, they are the primary focus of this thesis. Gyro resolution, sensitivity and non-

linearity have been analyzed extensively in public literature and therefore, are not covered

as part of this thesis.

Good design practice dictates that designers have estimates of expected non-idealities

before resorting to simulation tools for more detailed results. The primary goal of this the-

sis is to provide gyroscope designers with techniques for hand analysis of non-idealities.

Behavioral modeling and simulation is used throughout this thesis as a tool to verify hand

analysis as well as to provide quantitative data. Development of behavioral models and

solution techniques for associated simulation problems comprise a significant portion of

this thesis. In order to obtain a quantitative understanding of elastic and electrostatic cou-

pling in a CMOS-MEMS gyroscope, the following issues are discussed en route to the

gyroscope:

• General theory of in-plane elastic cross-axis coupling

• Out-of-plane elastic coupling in CMOS-MEMS beams

• Lateral and vertical curling of CMOS-MEMS beams with arbitrary boundary condi-

tions

5

• Model-order reduction for springs

• 3D modeling of electrostatic combs in the CMOS-MEMS process

• Convergence problems in MEMS behavioral simulations

In the next section, the organization of the topics listed above is described.

1.4 Thesis OrganizationThe thesis is organized as follows. Chapter 2 briefly surveys the development of

micromachined gyroscopes, introduces the CMOS-MEMS process and presents an intro-

duction to the behavioral simulation framework which is used extensively and also con-

tributed to in this thesis. Chapter 3 addresses elastic cross-coupling and thermoelastic

analysis for a restricted class of spring suspensions. Chapter 4 discusses reduced-order

modeling primarily of suspension beams elastic properties, but also suggests possible

extension to include viscous and inertial effects. Chapter 5 describes the electrostatic mod-

eling approach for CMOS-MEMS combs. Convergence problems in MEMS behavioral

simulation and guidelines for minimizing them are detailed in Chapter 6. Analysis and

simulation of non-idealities in the CMOS-MEMS gyroscope are presented in Chapter 7.

Finally, the contributions of the thesis are summarized in Chapter 8 and future directions

of work are suggested.

6

Chapter 2. Background

2.1 IntroductionMuch of the work in this thesis falls in the intersection of three complementary

research areas: micromachined gyroscopes, CMOS-MEMS and behavioral modeling and

simulation. Each of these areas is briefly reviewed in this chapter in order to place the

remainder of the thesis in perspective. In the initial section of this chapter, the develop-

ment of surface-micromachined gyroscopes, and, specifically, vertical-axis gyroscopes,

over the last decade is reviewed. Following this, a qualitative comparison between a repre-

sentative single-layer vertical-axis microgyroscope and a CMOS-MEMS vertical axis

gyroscope is made [10]. The CMOS-MEMS process and relevant non-ideal manufactur-

ing effects are then described. In the subsequent part of this chapter, the behavioral simu-

lation framework called Nodal Simulation of Sensors and Actuators (NODAS) [11][12],

which has been developed at Carnegie Mellon will be described. NODAS is used for micr-

ogyroscope simulation in this thesis. Additionally, models developed as part of this thesis

have been incorporated into NODAS.

2.2 Micromachined Gyroscopes2.2.1 Brief History

Microgyroscopes can be classified by a number of different criteria: by the manufac-

turing process into surface and bulk micromachined, in terms of the sensing axis as verti-

cal axis and lateral axis or in terms of the intended application range as rate grade, tactical

grade and inertial grade [2]. Most of the surface-micromachined gyroscopes reported so

far fall in the rate-grade category. The first microgyro reported in 1991 was a surface-

micromachined lateral axis gyroscope [13] followed up in [14]. Alternate microgyros built

using alternate sensing techniques: piezoresistive [15], tunneling-based [16] and optical

sensors [17] have also been reported. The first surface-micromachined vertical (Z) axis

gyroscope was made at the University of Michigan in 1994 [18]. This gyro used a vibrat-

ing ring suspended by radial springs to sense the Coriolis force. Most of the vertical-axis

microgyros developed since then are single-layer structures and use translational drive and

7

sense modes, in the plane of the structure [19][20][21][22][23][24][25][26][27][28][29].

Lateral [30][31] and vertical axis gyroscopes [10][32] have been built and successfully

tested in the multi-layer CMOS-MEMS process. In the next sub-section common features

of many single-layer vertical axis microgyroscopes are highlighted and compared with the

CMOS-MEMS vertical axis microgyro.

2.2.2 Common Features of Surface-Micromachined Vertical Axis Gyroscopes

The polysilicon microgyroscope developed by Clark et al. [19] at Berkeley is repre-

sentative of a number of later microgyroscope designs. As shown in Figure 2.1(a), linear

combs are used to actuate the inner-mass in the x direction. The inner plate vibrates with

large amplitude (few µm) in the x direction. The outer frame, along with the inner plate is,

free to oscillate in the y direction. A pair of differential combs on the outside are used to

pickup the Coriolis force induced vibrations in the y axis.

The outer frame is suspended by springs which are stiff in the x direction and there-

fore, has only a small amount of drive motion (few nm). Therefore, the Coriolis force due

to the vibration of the outer frame is insignificant. The Coriolis substantially acts only on

the central plate, but is transmitted to the rigid frame through the connecting beams which

are stiff in the y direction. Thus, only a fraction of the total mass available is being used to

sense the Coriolis force. Furthermore, it is seen that the central plate along with the rigid

frame is easily displaced in the y direction due to external accelerations. This opens up a

FIGURE 2.1. Topology comparison of (a) single-layer surface micromachinedgyroscopes (e.g., Clark et al. [19]) and (b) CMOS-MEMS nested gyroscope[10]. The dark shaded combs are the drive combs. Shading in the sense combsindicates different potentials.

x

y

(drive)

(sen

se)

input

φz

Driv

e co

mb

Driv

e co

mb

sense combs

(a) (b)

sense combs

drive combs

8

possibility of external accelerations coupling through to the output. There are alternate

suspension schemes, which completely decouple the sense and the drive modes

[26][32][33][34], but use linear combs for sensing purposes. Linear combs are less sensi-

tive than differential combs for same number of fingers.

Note that since the entire movable structure is at the same potential, the differential

sense fingers have to be anchored. There are two possible locations for the sense combs;

outside the movable frame as is shown in Figure 2.1(a) or inside the movable frame. The

suspension design is such that the drive motion cannot be decoupled from the inner plate.

Recall from Chapter 1 that the drive motion is more than 4 orders of magnitude larger than

the Coriolis force induced motion. Differential sense combs are typically non-ideal after

manufacturing. They can be expected to have a small sensitivity to cross-axis motions, as

will be shown in Chapter 7. Placing the differential sense combs inside the movable frame

will, therefore, lead to a significant sense signal due to the drive motion coupled to the

sense combs. Thus, in case of the single-layer microgyro with the suspension design as

shown, the only reasonable alternative is to place the differential combs outside the rigid

frame.

In contrast to the single-layer microgyro described above, the CMOS-MEMS nested

gyro topology [10] allows use of differential comb for sensing and, at the same time,

allows for decoupling of the drive and sense modes. In the next sub-section the vertical

axis CMOS-MEMS gyroscope which is used throughout this thesis for simulations is

described.

2.2.3 Vertical Axis CMOS-MEMS Gyroscope

The SEM of a nested gyroscope [10] is shown in Figure 2.2(a). This gyroscope is fab-

ricated in the CMOS-MEMS process [7]. It consists of an inner accelerometer nested

inside an outer resonator [10] as shown in Figure 2.2(b). The outer resonator is suspended

by four springs which are relatively rigid along the sensing direction (x) and compliant

along the driven direction (y). The outer resonator is driven at resonance and the inner res-

onator is forced to move along with the outer resonator because the springs suspending the

inner resonator are relatively rigid in y and compliant in x. In the presence of an angular

9

rate about the out-of-plane axis, both the resonators experience the Coriolis force in x,

however, the inner resonator has a larger displacement. The sensing mode resonant fre-

quency is designed to be larger than the drive frequency. The relative displacement

between the two resonators is sensed capacitively using differential combs.

Thus, the CMOS-MEMS gyroscope uses springs to decouple the drive and sense

modes and multiple conductors to place the differential sensing combs between the central

plate and the rigid frame, both of which are movable.

2.3 CMU CMOS-MEMS ProcessIn the CMOS-MEMS process developed at CMU [7][31][35], released microstruc-

tures are produced by two step post processing of a standard CMOS die. First, an anisos-

FIGURE 2.2. (a) SEM of the vertical axis CMOS-MEMS nested-gyroscope [10]. (b) Functionallyequivalent structure showing the inner accelerometer, outer rigid frame, inner and outer springsand drive and sense combs.

Drive comb

Oscillation sense-feedback comb Top sense comb

Bottom sense comb

x

y

(sense)

(dri

ve)

input rotation

Innerspring

Drive comb

Drive comb

sense combs

(a) CMOS-MEMS Gyroscope SEM

(b) Nested gyroscope functional equivalent

Inner accelerometerOuter rigid frame

Ωz

10

tropic reactive-ion etch (RIE) of the dielectric not covered by any metal layer is used to

define the sidewalls of the microstructures (Figure 2.3(b)). Following this, an isotropic

etch of the Silicon substrate leads to release of the suspended microstructures (Figure

2.3(c)).

The suspended microstructures are composed of a sandwich of metal and dielectric

layers. Since the materials have different thermal coefficients of expansion, the micro-

structures behave like thermal multi-morphs. Therefore, after processing, when the wafer

temperature is reduced to room temperature, residual stresses appear which tend to curl

the microstructure. In sense combs, vertical curling of the fingers leads to reduced sensi-

tivity because of reduced overlap area. The actuation force in case of driving combs is

degraded because of reduced change in capacitance with displacement. Furthermore,

(a)

(b)

(c)

CMOS circuits

silicon substrate

dielectric

polysiliconmicrostructures

metal-3metal-2

metal-1

anchoredstator

microstructureSF6-O2 Isotropic Etch beam

CHF3-O2 Anisotropic Etch

FIGURE 2.3. Abbreviated process flow for post-CMOS micromachiningdeveloped at CMU [7][31][35] (a) CMOS wafer cross-section with circuits andinterconnects (soon to be microstructures) (b) Oxide removal step (c)Microstructure release by Silicon removal

11

t

fringe capacitance due to the edges and corners of the comb-fingers becomes significant at

lower overlap areas. Curl-matching frames around the sensor have been proposed in order

to reduce the mismatch of the comb-fingers [36]. However, design of these frames for the

gyroscope under consideration is more complicated than the accelerometer in [36] and the

resultant curl is therefore, not as well-matched. Though vertical curling is seen throughout

the gyroscope it is the comb-drive which is affected significantly because it requires max-

imum overlap of the comb-fingers. Curling of the rest of the structure can be encapsulated

into a vertical displacement offset for the comb-drive and can be modeled by considering

different vertical positions of the comb-drive.

The CMOS-MEMS beams have embedded metal layers. Misalignment of the metal

layer mask during processing [38] can result in the metal layers inside the beam being off-

set from the center of the beam leading to an asymmetrical beam cross-section. This in

turn leads to elastic coupling between the in-plane and the out-of-plane modes and lateral

curling of beams and comb-fingers (Figure 2.4(a), (b)). Elastic coupling can lead to an

input offset in the microgyroscope due to the drive mode coupling onto the sense mode.

Geometrical offsets are caused by lateral curl of the fingers or the beams in the springs

(see Figure 2.5). As will be seen later, geometrical offsets give rise to input offsets and

cross-axis sensitivity. Other reasons for geometrical offsets include stress gradients along

FIGURE 2.4. (a) Lateral curling seen in beams with deliberately misaligned metal layout (b)Cross-section of the beam. It is seen that the METAL3 is not aligned with the METAL2 andMETAL1. (Pictures courtesy Xu Zhu and Hasnain Lakdawala)

(a) (b)

Lateral curling

MisalignmentIon milling effec

METAL3

METAL2

METAL1

12

the chip substrate or along the beam width. However, currently there are no available mea-

surements for stress gradients in the substrate or across the beam width.

Modeling approaches for the non-ideal manufacturing effects described above are

presented in the following chapters.

2.4 Structured Design Methodology for MEMSThere are on-going efforts to establish a hierarchy of design levels for MEMS

[11][12] similar to that existing in the digital design world. The basis for the hierarchy is

decomposition of MEMS devices into MEMS atomic elements such as plate masses,

beam springs, electrostatic gaps and anchors which are at a similar level as resistors,

capacitors and inductors in the electronics design hierarchy. This level is referred to as the

atomic level representation. An atomic level schematic representation of MEMS bears a

strong correspondence to the underlying layout.

At higher design levels, a chain of beam springs can be combined to form crab-leg

springs, u-shaped springs or serpentine springs. At an even higher (functional) level, all

the springs which connect two rigid elements (for instance, a plate and an anchor) can be

lumped together into a single functional spring element. The building blocks at this level

are “functional” elements such as mass, spring, damper, electrostatic sensor, electrostatic

actuator and differential sensor. Each of the functional elements exhibits only one kind of

functionality as opposed to the circuit-level atomic elements which incorporate multi-

domain physics. At the functional level, the different performance contributions are segre-

gated, requiring the parasitic-physics effects (i.e., mass of beams, damping forces on

FIGURE 2.5. (a) Geometrical offset in a differential comb-drive used in a CMOS accelerometer.One of the gaps is smaller than the other one (b) Laterally curled springs in an accelerometer(Pictures courtesy Vishal Gupta)

Non-identical gaps

Laterally curled springs

13

plates, finite stiffness of plates) to be computed and included in the appropriate functional

element. The functional level representation cannot be visually correlated to the layout of

the device. However, this level closely approximates the spring-mass-damper abstraction

of an inertial sensor, which designers use extensively in developing inertial sensors.

Abstracting away even the functional composition of the MEMS device, the macromodel

level, i.e., simply an equation summarizing the input-output relationship of the device, is

obtained.

The MEMS design hierarchy is summarized in Figure 2.6 which shows the layout

level, atomic level schematic, the functional schematic and the macromodel representa-

tion. A design hierarchy is not of use unless the different levels of the hierarchy can be tra-

versed with ease. Broadly, upward motion through the hierarchy, leading to increasing

abstraction, is referred to as extraction or verification. Downward motion, resulting in

increased visibility of finer details, is called synthesis. Over the past decade, several

research efforts, notably at CMU and other universities as well, have not only developed

hierarchical representations of MEMS but also demonstrated automated methodologies

for various components of the hierarchy traversal.

The NODAS framework, developed at CMU, implements the hierarchical representa-

tion of MEMS described above. Schematics of MEMS sensors are created using parame-

FIGURE 2.6. MEMS design hierarchy

Vo SΩz=

Layout

Atomic schematic

Functional schematic

Macromodel

M

K B

F

Extraction/Verification

Synthesis

14

terized elements such as beams, plates, anchors and combs and electrical and mechanical

independent sources. DC, AC and transient analysis can then be performed on the MEMS

sensors. Behavioral simulation of the CMOS-MEMS vertical axis microgyroscope using

atomic level elements is used in this thesis to verify hand analysis and to quantify non-

ideal effects. The Spectre circuit simulator from Cadence is used as the simulation engine

for the behavioral simulations. Modeling approaches for cross-axis coupling in springs,

vertical curling and mask misalignment in beams and multi-layer effects in combs are

described in the following chapters. The resultant improved models are incorporated into

the NODAS library. One of the fundamental tasks which provides the back-bone for such

a structured design methodology is building behavioral models. A brief summary of mod-

eling approaches is given in the following sub-section.

2.4.1 Modeling

In the context of the design hierarchy mentioned above, modeling can be viewed as a

process of relating parameters at a higher-level of the hierarchy to the parameters at a

lower-level. Availability of models which are accurate and can also be evaluated fast

enables easy traversal between the different levels possible along both, extraction and syn-

thesis directions. At the level of atomic-elements, modeling involves identification and

encoding of significant relationships between geometrical parameters and functional

parameters. Examples include derivation of equations for spring stiffness from beam geo-

metrical parameters (width, length, cross-section etc.) and plate mass and moments of

inertia from plate length, width and composition. The models implicitly assume a set of

manufacturing process-dependent constants for material properties. Those familiar with

models in the circuit world can immediately correlate this modeling procedure to the deri-

vation of transistor I-V relationships in terms of geometry and process-dependent doping

and material properties. At the same time, those familiar with modeling in the mechanical

world can distinguish this process from the building of “solid models” for use in numeri-

cal solvers and visualization. Modeling in elastic and electrostatic domains done in this

thesis is explained in detail in following chapters. In this section a brief overview of mod-

eling in elastic and electrostatic domains is presented.

15

2.4.2 Elastic Models

The elastic models in this thesis are based upon linear beam theory [39] wherein

forces and bending moments are linearly related to translational and rotational displace-

ments. Shear and non-linear effects are not being considered in this thesis, but are given

considerable attention in a parallel work [40]. Linear beam theory is based upon the fol-

lowing fundamental differential equation, which is valid when there are no distributed

loads, i.e., forces and moments are applied only at the two end-points of a beam [39]:

(2.1)

where, is the location along the length of the beam and is the displacement along one

of the two orthogonal directions as shown in Figure 2.7. Energy methods, described in

detail in [41], are used to derive equations for spring stiffnesses. A brief introduction to

energy methods is given below by way of deriving the compliance matrix for a single

beam which is part of a spring.

A number of common spring topologies such as crab-leg, u-shaped and serpentine

springs belong to a larger class, in which each spring is a single chain of beams. The ana-

lytical advantage in dealing with this class of springs is due to the fact that the forces

transmitted through the beams remain invariant from the load point to the anchor point.

Figure 2.8 shows a spring composed of 9 beams in a single-chain configuration, attached

to a rigid plate at one end and anchored at the other end. The procedure for computing the

in-plane compliance matrix for a single beam in the spring is described below. A force (or

moment) is applied to the point C, in the direction of interest, and the displacement is cal-

culated symbolically (as a function of the design variables and the applied force). When

x4

4

d

d y 0=

x y

FIGURE 2.7. The direction along the length (x) and the direction of deflection ofa beam (y)

Anchor

x

yUndeflected Beam

Deflected beam

16

forces (moments) are applied at the end-points of the flexure, assuming linear beam the-

ory, the energy per unit length of the beam is given as:

(2.2)

The total energy of deformation, U, is obtained by integrating over the length of the beam

followed by summation over all the beams:

(2.3)

where, Li is the length of the i’th beam in the flexure, is the bending moment trans-

mitted through beam i, E is the Young’s modulus of the structural material and Ii is the

moment of inertia of beam i, about the relevant axis (z axis for in-plane forces and

moments about z). The bending moment is a linear function of the forces and moments

applied to the end-points of the flexure. Furthermore, it varies linearly with the position

along the length of the beam. Therefore, the energy stored in the beam due to displacement

is quadratically dependent on the applied forces and moments. In particular, for a single

chain of beams (Figure 2.8(a)), the bending moment and, therefore, the energy stored in a

beam, depends only on the position of the end-points of the beam relative to the point of

application of force C. The displacement of point C in any direction ζ is given as:

(2.4)

where, Fζ is the force applied in that direction [39]. Similarly, angular displacements can

be related to applied moments. The moment being linearly dependent on the

applied forces and moments, the displacement is also a linear function of the applied

forces, i.e.,

(2.5)

xddU Mi x( )

2

2EIi-----------------=

UMi x( )

2

2EIi----------------- xd

0

Li∫

beam i 1=

N

∑=

Mi x( )

δζ ∂U∂Fζ---------=

Mi x( )

δζi αζξiFξiξ∑=

17

where is a generalized displacement (includes translation and rotation), is the

generalized force in the direction and , the compliance of the ith beam.

The in-plane compliance matrix for a beam, derived in terms of the end-points of the

beam, is given as:

(2.6)

where,

C123

45

6

7 8 9

FIGURE 2.8. (a) Spring with single-chain of 9 beams attached to a plate. C is thepoint of application of force. The other end of the spring is anchored. (b) Free-body diagram of beam 6 and the bending moment along beam 6.

anch

or

plateFx

Fy Mz

-Fx

-Fy

Fx

Fy

(a) (b)

Mz Fy x16 xC–( ) Fx y16 yC–( )+–( )

Mz Fy x26 xC–( ) Fx y26 yC–( )+–( )–

x16 y16,( )

x26 y26,( )

δζi Fξi

ξ αζξi

αxxi αxyi αxφzi

αyxi αyyi αyφzi

αφzxiαφzyi

αφzφzi

αxxili y1i

2 y2i2 y1i y2i 3yC–( ) 3y2iyC– 3y2iyC– 3yC

2++ +( )3EIzi

------------------------------------------------------------------------------------------------------------------------------------=

αyyili x1i

2 x2i2 x1i x2i 3xC–( ) 3x2ixC– 3x2ixC– 3xC

2++ +( )3EIzi

------------------------------------------------------------------------------------------------------------------------------------=

αφzφzili

EIzi---------=

αxyi αyxili x1i 2y1i y2i 3yC–+( ) x2i y1i 2y2i 3yC–+( ) 3xC y1i y2i 2yC–+( )–+( )

6EIzi-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------–= =

αxφziαφzxi

li y1 i y2 i 2yC–+( )2EIzi

--------------------------------------------= =

18

The elements of the in-plane compliance matrix derived above are used in this thesis to

derive properties of symmetric springs as well as to compute stiffness values for an entire

spring.

2.4.3 Electrostatic Modeling

Electrostatic modeling for gyroscopes mainly involves deriving geometry-based

equations for capacitance and force between two or more electrodes in combs, parallel

plates and other kinds of sensing or actuation structures. Fundamentally, deriving equa-

tions for capacitance involves solving the Laplace equation with appropriate boundary

conditions:

(2.7)

where, is the electrostatic potential which is generally a function of spatial loca-

tion. Instead of solving the Laplace equation for an entire sensor or actuator, which is sel-

dom practical, usually symmetry considerations are used to break up the sensor into a

number of smaller structures which can be solved much more easily. The total energy

which is stored in a capacitor with a voltage V applied between the two plates is given as:

(2.8)

Once the capacitance has been derived as a function of the relative displacement

between the two electrodes the force can be obtained by using the principle of virtual work

and differentiating the total energy of the system (if the capacitance is independent of the

voltage). For example, the force along the x direction will be given as: